Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+33373x+3451358\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+33373xz^2+3451358z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+43252029x+160896814398\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(24, 2053)$ | $0$ | $3$ |
Integral points
\( \left(24, 2053\right) \), \( \left(24, -2078\right) \)
Invariants
| Conductor: | $N$ | = | \( 12138 \) | = | $2 \cdot 3 \cdot 7 \cdot 17^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-7516584706022784$ | = | $-1 \cdot 2^{7} \cdot 3^{15} \cdot 7^{2} \cdot 17^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{49218965184023}{89996344704} \) | = | $2^{-7} \cdot 3^{-15} \cdot 7^{-2} \cdot 17^{2} \cdot 23^{3} \cdot 241^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7310631048445564575130084990$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.78665865682581776409649695971$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0426921261059747$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.639477167136787$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.28691314284528674136656221017$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 90 $ = $ 1\cdot( 3 \cdot 5 )\cdot2\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.8691314284528674136656221018 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.869131428 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.286913 \cdot 1.000000 \cdot 90}{3^2} \\ & \approx 2.869131428\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 90720 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $17$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-24.d.1.8, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 1 \\ 19 & 6 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 867 = 3 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 4046 = 2 \cdot 7 \cdot 17^{2} \) |
| $5$ | good | $2$ | \( 4046 = 2 \cdot 7 \cdot 17^{2} \) |
| $7$ | split multiplicative | $8$ | \( 867 = 3 \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 42 = 2 \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 12138.p
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.6936.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.1154594304.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.86630653872.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.13305294502748580528.2 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.10652087284684014119104200037348933632.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 17 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | split | ord | split | add |
| $\lambda$-invariant(s) | 3 | 3 | 2 | 1 | - |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.